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Long time behavior of a tumorimmune system competition model perturbed by environmental noise
Advances in Difference Equations volume 2017, Article number: 58 (2017)
Abstract
This paper investigates the long time behavior of tumor cells evolution in a tumorimmune system competition model perturbed by environmental noise. Sufficient conditions for extinction, stochastic persistence, and strong persistence in the mean of tumor cells are derived by constructing Lyapunov functions. The study results show that environmental noise can accelerate the extinction of tumor cells under immune surveillance of effector cells, which means that noise is favorable for the extinction of tumor in this condition. Finally, numerical simulations are introduced to support our results.
Introduction
Cancer is becoming the leading cause of death around the world, but our cognition of its causes, methods of prevention and cure are still in its infancy. One great method that has shown its potential in our better understanding of such a complicated biological problem is mathematical modeling [1]. Tumor immune models have existed since the early 1990s. Researchers have proposed various modeling approaches using ordinary and delayed differential equations [2–5]. A good summary of early works of tumor immune dynamics can be found in [6]. A detailed review of nonspatial models described by ODEs is published by Eftimie [7]. A simple but classical mathematical model of a cell mediated response to a growing tumor cell is proposed and analyzed by Kuznetsov and Taylor [8]; it takes into account the penetration of tumor cells by effector cells as well as the inactivation of effector cells. Their model can be applied to describe two different mechanisms of the tumor: tumor dormancy and sneaking through. Galach [9] firstly simplified the KuznetsovTaylor model by replacing the MichaelisMenten form with a LotkaVolterra form for the immune reactions. Then time delay was considered in the simplified model, and a state of the returning tumor was observed. More complete bibliography about the evolution of a cell and the relevant role of cellular phenomena in directing the body toward recovery or toward illness can be found in [10, 11]. The detailed descriptions of virus, antivirus, and body dynamics are available in [12–15].
Conventional treatments of cancer, such as radiotherapy and chemotherapy, usually act on the tumor cells themselves, and although these conventional treatments respond early, recurrences and drug resistance often occur late in the course of longterm treatment. Cancer immunotherapy has recently gained exciting progress. In contrast to conventional therapies, immunotherapy elicits immune system antitumor responses by acting on the immune system. ImmuneCheckpoint Inhibitors clinical trials have shown a greater sustained response than conventional chemotherapy. The positive response of immunotherapy generally depends on the interaction of tumor cells and the immune regulation in the tumor microenvironment. In fact, tumor microenvironment is inevitably affected by environmental noise, which is an important component in realism. Nowadays, noise dynamics have been widely studied in different fields such as epidemic model [16], nervous system [17], genetic regulatory system [18–20], bistable system [21–23], chaotic system [24, 25], etc. In the last years, stochastic growth models for cancer cells were studied in [26–30], in which Lyapunov exponent method and FokkerPlanck equation method are used to investigate the stability of the stochastic model by numerical simulations. The goal of this paper is to explore the long time behavior of tumor and effector cells in the tumorimmune system competition model perturbed by environmental noise. One of many advantages of our paper is that we initially make use of the methods of Itô’s formula and Lyapunov function to derive and analyze the properties of a stochastic tumorimmune system competition model. The other advantage of this paper is that the conditions for extinction, stochastic persistence, and strong persistence in the mean of tumor cells are established by strict mathematical proofs. Accordingly, the sufficient conditions for extinction and persistence could provide us more effective and precise therapeutic schedule to eliminate tumor cells and improve the treatment of cancer.
This paper is organized as follows. In Section 2, we introduce a mathematical model. In Section 3, we establish the sufficient conditions for extinction, stochastic persistence, and strong persistence in the mean of tumor cells. Numerical simulations are presented in Section 4, and they are used to verify and illustrate the theorems of Section 3. In Section 5, we discuss the conclusions and future directions of the research.
Mathematical model
When unknown tissues, organisms, or tumor cells appear in a body, the immune system tries to identify them and, if it succeeds, it attempts to eliminate them. The immune system response consists of two distinct interacting responses: the cellular response and the humoral response. The cellular response is carried by T lymphocytes. The humoral response is mediated by B lymphocytes. The dynamics of the antitumor immune response in vivo is complicated and not well understood.
Once the tumor cells are identified, the immune response begins. Then tumor cells are caught by macrophages, which can be found in all tissues in the body and circulate in the blood stream. Macrophages absorb tumor cells, eat them, and release a series of cytokines which activate T helper cells (i.e., a subpopulation of T lymphocytes). Activated T helper cells coordinate the counterattack. T helper cells can also be directly stimulated to interact with antigens. These helper cells cannot kill tumor cells, but they send urgent biochemical signals to a special type of T lymphocytes called natural killers (NKs). T cells begin to multiply and release other cytokines that further stimulate more T cells, B cells, and NK cells. As the number of B cells increases, T helper cells send a signal to start the production of antibodies. Antibodies circulate in the blood and are attached to tumor cells, which implies that the tumor cells are more quickly engulfed by macrophages or killed by NK cells. Like all T cells, NK cells are programmed to identify one specific type of infected cell or cancer cell. NK cells are lethal and constitute a vital line of the defense.
The model presented in [8] describes the response of effector cells (ECs) to the growth of tumor cells (TCs). This model differs from others because it takes into account the penetration of TCs by ECs, which simultaneously causes the inactivation of ECs. It is assumed that interactions between ECs and TCs in vitro can be described by the kinetic scheme shown in Figure 1, where E, T, C, \(E^{\ast }\), and \(T^{\ast }\) are the local concentrations of ECs, TCs, ECTC complexes, inactivated ECs, and lethally hit TCs, respectively. \(k_{1}\) and \(k_{1}\) denote the rates of bindings of ECs to TCs and the detachment of ECs from TCs without damaging TCs, \(k_{2}\) is the rate at which ECTC interactions program TCs for lysis, and \(k_{3}\) is the rate at which ECTC interactions inactivate ECs.
Kuznetsov and Taylor’s model [8] is as follows:
where s is the normal (i.e., not increased by the presence of the tumor) rate of the flow of adult ECs into the tumor site. \(F(C,T)\) describes the accumulation of ECs in the tumor site; \(d_{1}\), \(d_{2}\), and \(d_{3}\) are the coefficients of the processes of destruction and migration for E, \(E^{\ast }\), and \(T^{\ast }\), respectively; a is the coefficient of the maximal growth of tumor; and \(1/b\) is the environment capacity.
It is claimed in [8] that experimental observations motivate the approximation \(\frac{dC}{dt}\approx 0\). Therefore, it is assumed that \(C\approx {KET}\), where \(K= \frac{k_{1}}{k_{2}+k_{3}+k_{1}}\), and the model can be reduced to two equations which describe the behavior of ECs and TCs only. Moreover, in [9] it is suggested that the function F should be in the form \(F(C,T) = F(E,T) = \theta ET\). Therefore, model (1) takes the form
where \(\alpha_{1}=\theta m\), and a, b, s have the same meanings as those in (1); \(n= Kk_{2}\), \(m = Kk_{3}\), and \(d = d_{1}\), respectively. All coefficients except \(\alpha_{1}\) are positive. \(\alpha_{1}\) is the immune response to the appearance of the tumor cells, its sign depends on the relation between θ and m. If the stimulation coefficient of the immune system exceeds the neutralization coefficient of ECs during the formation of ECTC complexes, then \(\alpha_{1}> 0\). We use the dimensionless form of model (2),
where x denotes the dimensionless density of ECs; y stands for the dimensionless density of the population of TCs; \(E_{0}\) and \(T_{0}\) are the initial value. \(\alpha =\frac{a}{nT_{0}}\), \(\beta =bT _{0}\), \(\delta =\frac{d}{nT_{0}}\), \(\varepsilon = \frac{s}{nE_{0}T_{0}}\), and \(\omega =\frac{\alpha_{1}}{n}\) represent the immune response to the appearance of the tumor cells (i.e., immune coefficient). In this paper we consider the case of \(\omega >0\), which means that immune response is positive.
In [9], system (3) always has the equilibrium \(P_{0}=(\frac{\varepsilon }{\delta },0)\). If \(\omega >0\) and \(\alpha \delta <\varepsilon \), then \(P_{0}\) is the unique equilibrium of (3) and it is globally stable. If \(\omega >0\) and \(\alpha \delta >\varepsilon \), then \(P_{0}\) is unstable, and there is an equilibrium
of (3), which is globally stable, and \(\Delta =\alpha^{2}( \beta \delta \omega)^{2}+4\alpha \beta \varepsilon \omega \).
In fact, tumor microenvironment is complex, and the growth of tumor cells is influenced by many environmental factors [31], e.g., the supply of oxygen and nutrients, the degree of vascularization of tissues, the immunological state of the host, chemical agents, temperature, etc. A tiny variation of parameters will change the evolutionary process of tumor cells. So, it is inevitable to consider the tumorimmune system competition model with environmental noise. In this paper, taking into account the effect of randomly fluctuating environment, we assume that fluctuations in the environment will mainly affect the immune coefficient ω
where \(B(t)\) is standard Brownian motions with \(B(0)=0\), and with the intensity of white noise \(\sigma^{2}>0\). The stochastic version corresponding to the deterministic model (3) takes the following form:
where all the parameters are positive and bounded. For convenience of expression, we define the following notions:
Long time behavior of the stochastic model
Our primary interests in tumor dynamics are the extinction and survival of tumors induced by environmental fluctuations. In order to study the extinction and survival, we need some appropriate definitions about extinction and persistence. Here we adopt the concepts of extinction, stochastically permanent [32], and strong persistence in the mean [33]. In addition, some of our proofs are motivated by the works Liu [33], Mao [34], and Zhao [16]. The useful definitions are as follows:

(1)
The tumor cells \(y(t)\) will go to extinction a.s. if \(\lim_{t\rightarrow +\infty }y(t)=0\).

(2)
The tumor cells \(y(t)\) will be stochastically permanent a.s. if there are constants \(N > 0\) and \(M > 0\) such that \(\mathcal{P}_{ \ast }\{y(t)\geq N\}\geq 1\xi \) and \(\mathcal{P}_{\ast }\{y(t) \leq M\}\geq 1\xi\).

(3)
The tumor cells \(y(t)\) will be strong persistent in the mean a.s. if \({\langle y(t)\rangle }_{\ast }>0\).
Next we establish the sufficient conditions of extinction and persistence for our model.
Lemma 1
For any positive initial value \((x_{0},y_{0})\), if \(0< x_{0}<\frac{1}{ \beta }\), the solution of (4) obeys
Proof
According to the second equation of model (4), we have
Firstly, we discuss y based on the different value range of x.

(i)
When \(0<\alpha x\) and \(\alpha \beta y_{0}\leq \alpha x\) (i.e., \(0< x\leq \alpha (1\beta y_{0})\)),
$$\begin{aligned} y =&\frac{1}{{{\frac{\alpha \beta }{\alpha x}+(\frac{1}{y_{0}}\frac{ \alpha \beta }{\alpha x}})e^{(\alpha x)t}}} \\ =&\frac{1}{\frac{\alpha \beta }{\alpha x}+\frac{\alpha \beta }{ \alpha x}(\frac{\alpha x}{\alpha \beta y_{0}}1)e^{(\alpha x)t}} \\ \leq& \frac{1}{\frac{\alpha \beta }{\alpha x}} \\ \leq& \frac{1}{ \beta }. \end{aligned}$$ 
(ii)
When \(0<\alpha x<\alpha \beta y_{0}\) (i.e., \(\alpha (1\beta y_{0})< x<\alpha \)),
$$\begin{aligned} y =&\frac{\alpha x}{\alpha \beta +(\frac{\alpha x}{y_{0}}\alpha \beta)e^{(\alpha x)t}} \\ < &\frac{\alpha x}{{\alpha \beta +(\frac{ \alpha x}{y_{0}}\alpha \beta)}}=y_{0}. \end{aligned}$$ 
(iii)
When \(\alpha x\leq 0\) (i.e., \(x\geq \alpha \)),
$$\begin{aligned} y &=\frac{x\alpha }{\alpha \beta +(\alpha \beta \frac{\alpha x}{y _{0}})e^{(\alpha x)t}} \\ &=\frac{x\alpha }{\alpha \beta +(\alpha \beta +\frac{x\alpha }{y _{0}})e^{(\alpha x)t}} \\ &\leq \frac{x\alpha }{\alpha \beta + \alpha \beta +\frac{x\alpha }{y_{0}}}=y_{0}. \end{aligned}$$
Consequently, we have proved \(y(t)\leq \max \{y_{0}, \frac{1}{\beta } \}\). Then we will show that \(x(t)\) is bounded. Applying Itô’s formula [35] to the first equation of model (4), we get
Integrating both sides from 0 to t, we can obtain
where \(M_{1}(s)=\frac{\beta (1\beta x(s))(\varepsilon +\omega x(s)y(s) \delta x(s))+0.5\beta^{2}\sigma^{2} x^{2}(s) y^{2}(s)}{(\beta x(s)1)^{2}}\), and \(M_{2}(s)= \frac{\beta \sigma x(s)y(s)}{\beta x(s)1}\).
Therefore,
That is to say, if \(x_{0}<\frac{1}{\beta }\), then \(x(t)<\frac{1}{ \beta } \) for all \(t>0\). □
Lemma 2
Let \(f \in C[[0,\infty)\times \Omega,(0,\infty)]\) and \(F(t)\in C([0, \infty)\times \Omega,R)\). If there exist positive constants \(\lambda_{0}\), λ, and T such that
for all \(t\geq T\), and \(\lim_{t\rightarrow \infty }\frac{F(t)}{t}=0\) a.s., then
Proof
The proof is similar to the proof of Lemma in [16]. Note that \(\lim_{t\rightarrow \infty }\frac{F(t)}{t}=0\) a.s.; then, for arbitrary \(k>0\), there exist \(T_{0}=T_{0}(\omega)>0\) and a set \(\Omega_{k}\) such that \(P(\Omega_{k})\geq 1k\) and \(\frac{F(t)}{t} \leq k\) for all \(t\geq T_{0}\), \(\omega \in \Omega_{k}\). Let \(\bar{T}=\max \{T,T_{0}\}\) and
Since \(f \in C[[0,\infty)\times \Omega,(0,\infty)]\), then \(\varphi (t)\) is differentiable on \([\bar{T},\infty)\) a.s. and
Substituting \(\frac{d\varphi (t)}{dt}\) and \(\varphi (t)\) into (7), we obtain
Thus
Integrating this inequality from T̄ to t results in
This inequality can be rewritten as
Taking the logarithm on both sides yields
where
or
Dividing both sides by \(t(\geq \bar{T}>0)\) gives
Taking the limit inferior on both sides and applying L’Hospital’s rule on the righthand side of this inequality, we obtain
Letting \(k\rightarrow 0\) yields
□
Theorem 1
For any positive initial value \((x_{0},y_{0})\), particularly, \(x_{0}<\frac{1}{\beta }\), Equation (4) has a positive unique global solution \((x(t),y(t))\) on \(t\geq 0\) a.s.
Proof
To get a unique global solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and the local Lipschitz condition [36]. However, the coefficients of model (4) do not satisfy the linear growth condition, so the solution may explode in a finite time. Since the coefficients of Equation (4) are locally Lipschitz continuous for any given initial value \((x_{0},y_{0})\in R _{+}^{2}\), there is a unique maximal local solution \((x(t),y(t))\) on \(t\in [0,\tau_{e}]\), where \(\tau_{e}\) is the explosion time [36]. To show this solution is global, we only need to show that \(\tau_{e}=\infty \). To this end, let \(k_{0}>0\) be sufficiently large so that \(x_{0}\), \(y_{0}\) all lie within the interval \([\frac{1}{k_{0}},k _{0}]\). For each integer \(k\geq k_{0}\), we define the stopping times \(\tau_{k}= \inf \{t\in [0,\tau_{e}]:\min \{x(t),y(t)\}\leq \frac{1}{k}\mbox{ or }\max \{x(t),y(t)\}\geq k\}\). Clearly, \(\tau_{k}\) is increasing as \(k\rightarrow \infty \). Set \(\tau_{\infty }= \lim_{k \rightarrow +\infty } \tau_{k}\), thus \(\tau_{\infty }\leq \tau _{e}\) a.s. In other words, we only need to prove \(\tau_{\infty }= \infty \). If this statement is false, there exist constants \(T>0\) and \(\varepsilon \in (0,1)\) so that \(\mathcal{P}\{\tau_{\infty }<\infty \}>\varepsilon \). Thus there is an integer \(k_{1}>k_{0}\) such that
Define a \(C^{2}\)function \(V:R_{+}^{2}\rightarrow R_{+}\) by \(V(x,y)=(x1\ln x)+(y1\ln y)\). The nonnegativity of this function can be seen from \(u1\ln u\geq 0\), \(\forall u>0\). Let \(k\geq k_{0}\) and \(T>0\) be arbitrary. Applying Itô’s formula, we have
Here, L is a positive constant, and in the proof of the last inequality, we have used Lemma 1 (i.e., for \(\forall t\geq 0\), \(x(t)\) and \(y(t)\) are bounded). The inequality in (9) implies that
Taking expectation on both sides of the above inequality, we can obtain
Let \(\Omega_{k}=\{\tau_{k}\wedge T\}\), then by Inequality (8) we have \(P(\Omega_{k})\geq \varepsilon \). Note that for any \(\omega \in \Omega_{k}\), \(x(\tau_{k},\omega)\), \(y(\tau_{k},\omega)\) equals either k or \(1/k\), hence \(V(x(\tau_{k},\omega),y(\tau_{k}, \omega))\) is no less than \(\min \{2(k1\ln k),2(\frac{1}{k}1+ \ln k),k+\frac{1}{k}2\}\). By Formula (10) we have
where \(1_{\Omega_{k}}\) is the indicator function of \(\Omega_{k}\). Letting \(k\rightarrow \infty \), the contradiction \(\infty >V(x_{0},y _{0})+LT=\infty \) exists, which completes the proof. □
Remark 1
In order to guarantee the existence and uniqueness of the solution of model (4), we discuss the extinction and persistence of \(y(t)\) under the condition \(x_{0}<\frac{1}{\beta }\) below.
Theorem 2
Let \((x(t), y(t))\) be the solution of system (4) with positive initial value \((x_{0},y_{0})\). If \(\alpha \delta <\varepsilon \), then
Proof
An integration of the first equation of model (4) yields
We compute that
where \(N_{1}(t)=\frac{\sigma }{\delta }\int_{0}^{t} x(s)y(s)\,dB(s)\), which is a local continuous martingale and \(N_{1}(0)=0\). Moreover,
By the strong law of large numbers for local martingales [36], we obtain
Taking the limit inferior on both sides of (12), we have
Applying Itô’s formula to the second equation of model (4), we get
Integrating this from 0 to t and dividing by t on both sides, we have
Taking the limit superior on both sides of (15) and substituting (14) into (15) yields
If the condition \(\alpha \delta <\varepsilon \) is satisfied, then
which implies
Applying Itô’s formula to the first equation of model (4) leads to
Integrating this from 0 to t, we have
Set \(N_{2}(t)=\int_{0}^{t} \sigma y(s)\,dB(s)\), whose quadratic variation is
By virtue of the exponential martingale inequality [36], for any positive constants T, a and b, we have
Choose \(T=n\), \(a=1\), \(b=2\ln n\), we get
An application of the BorelCantelli lemma [36] yields that for almost all \(\omega \in \Omega \), there is a random integer \(n_{0}=n_{0}(\omega)\) such that for \(n\geq n_{0}\),
That is to say,
for all \(0\leq t\leq n\), \(n\geq n_{0}\) a.s. Substituting the above inequality into (17) leads to
By use of Lemma 2, we have
Namely,
Together (18) with (14), we see that
Applying L’Hospital’s rule to the left side of (19), we can derive that
□
Theorem 3
If \(\alpha \delta \frac{\omega \alpha }{4\beta }>\varepsilon \), then the tumor cells \(y(t)\) will be stochastically permanent almost surely.
Proof
We first show that for arbitrary fixed \(\xi >0\), there exists a constant \(M>0\) such that \(\mathcal{P}_{\ast }(y(t)\leq M)\geq 1 \xi \). Define \(V(y)=y^{q} \) for \(y\in R_{+}\), where \(0< q<1\). Then it follows from Itô’s formula that
Let \(n_{0}\) be so large that \(y_{0}\in [1/n_{0},n_{0}]\). For each integer \(n\geq n_{0}\), define the stopping time \(\tau_{n}=\inf \{t \geq 0:y(t) \notin (1/n,n)\}\). Clearly, \(\tau_{n}\rightarrow \infty \) almost surely as \(n\rightarrow \infty \). Applying Itô’s formula again to \(\exp(t)V(y)\) gives
where K is a positive constant. Integrating this inequality and then taking expectations on both sides, one can see that
Letting \(n\rightarrow \infty \) yields \(\exp(t)E[y^{q}(t)]\leq y_{0} ^{q}+K(\exp(t)1)\), or
In other words, we have already shown that \(\limsup_{t\rightarrow +\infty }E[y^{q}(t)]\leq K\). Thus, for any given \(\xi >0\), let \(M=K^{1/q}/\xi^{1/q}\), by virtue of Chebyshev’s inequality, we can derive that
that is to say, \(\mathcal{P}^{\ast }\{y(t)>M\}\leq \xi \). Consequently, \(\mathcal{P}_{\ast }\{y(t)\leq M\}\geq 1\xi \).
Next we demonstrate that for arbitrary given \(\xi >0\), there exists a constant \(N>0\) such that \(\mathcal{P}_{\ast }(y(t)\geq N)\geq 1 \xi \). Define \(V_{1}(y)=y^{2}\) for \(y\in R_{+}\). Applying Itô’s formula to \(V_{1}(y)\) leads to
Define \(V_{2}(y)=(1+V_{1}(y))^{\theta }\), where \(\theta \in (1/4,1/2)\). Applying Itô’s formula again to \(V_{2}(y)\) gives
An integration of system (4) is
According to (22), we have
We compute that
Taking the limit superior on both sides of the above inequality yields
Applying L’Hospital’s rule to (24), we obtain \((x(t))^{\ast } \leq \frac{\varepsilon }{\delta }+\frac{\omega \alpha }{4\delta \beta }\). Letting \(r(t)=\alpha x(t)\), with the condition of Theorem 3, we can derive that \(r_{\ast }\geq \alpha (\frac{ \varepsilon }{\delta }+\frac{\omega \alpha }{4\delta \beta })>0\), applying this to (21) yields
for sufficiently large \(t\geq T\). Now, let \(\eta >0\) be sufficiently small to satisfy \(0<\eta /\theta <2(r_{\ast }\xi)\). Define \(V_{3}(y)=\exp(\eta t)V_{2}(y)\). Making use of Itô’s formula gives
for \(t\geq T\). Note that \(H(y)\) is upper bounded in \(R_{+}\), namely \(C_{1} :=\sup_{y\in R_{+}}H(y)<+\infty \). Consequently,
for sufficiently large t. Integrating both sides of the above inequality yields
Then taking expectations on both sides of the above inequality gives
We compute that
In other words, we have shown that
Thus, for any given \(\xi >0\), let \(N=\xi^{0.5/\theta }/C^{0.5/\theta }\), by Chebyshev’s inequality,
That is to say, \(\mathcal{P}^{\ast }\{y(t)< N\}\leq \xi \). Thus \(\mathcal{P}_{\ast }\{y(t)\geq N\}\geq 1\xi \). □
Theorem 4
If \(\delta \alpha >\varepsilon \), then the tumor cells \(y(t)\) will be strongly persistent in the mean almost surely.
Proof
According to (23), we compute that
Applying Itô’s formula to the second equation of system (4) leads to
Integrating this from 0 to t and dividing by t on both sides, we have
Substituting (25) to (26) yields
We compute that
Consider that both \(x(t)\) and \(y(t)\) are bounded, we have
Consequently, we can derive that if \(\delta \alpha >\varepsilon \), then \({\langle y(t)\rangle }_{*}>0\) a.s. □
Numerical simulations
In this section, we use the EulerMaruyama numerical method mentioned by Higham [37] to support our results.
Figure 2 shows simulations of the results in Theorem 2. Based on the condition of Theorem 2, we choose \(\varepsilon =110\), \(\delta =35\), \(\alpha =3.02\). Note that \(\alpha \delta =105.7<\varepsilon =110\), and \(\frac{\varepsilon }{ \delta }=3.143\). Figure 2 demonstrates that tumor cells will tend to zero, and effector cells will tend to the constant 3.143 over time.
Figure 3 presents simulations of the results in Theorems 3 and 4. In Figure 3(a), we choose \(\varepsilon =83\), \(\delta =35\), \(\alpha =3.02\). Then the conditions satisfy \(\delta \alpha \frac{\omega \alpha }{4\beta }=86.825>\varepsilon =83\). Applying Theorem 3, we can see that \(y(t)\) will be stochastically permanent. Figure 3(a) confirms the results. In Figure 3(b), we choose \(\varepsilon =100, \delta =35, \alpha =3.02\). That is to say, \(\alpha \delta =105.7>\varepsilon =100\). By virtue of Theorem 4, one can get that \(y(t)\) will be strongly persistent in the mean and \(\langle y(t)\rangle_{\ast }\geq \frac{\delta \alpha  \varepsilon }{\alpha \beta \delta +\omega \alpha }= 15.7285\).
Figure 4 shows the effects of environmental noise on the mean time to the extinction of tumorimmune system competition system. Firstly, in Figure 4(a), we keep the parameter value ε at 109, which represents the normal rates of the flow of adult ECs into the tumor site, and simulate the path of \(y(t)\) in both the stochastic model and its corresponding deterministic model. It is shown that \(y(t)\) in the stochastic model and the deterministic model tends to zero since \(\varepsilon =109>\alpha \delta =105.7\). However, the rate of tumor reduction in the stochastic model is faster than that in the deterministic model. These results can also be confirmed in Figure 4(b). Figure 4(b) simulates the mean time to extinction (MTE) as a function of ε, where the MTE is the average of \(1{,}000\) times of a sample path. It is obvious that the MTE in the stochastic model is less than that in the deterministic model when the values of ε are the same, which shows that environmental noise can accelerate the extinction of tumor cells. In addition, it can be seen in Figure 4(b) that the MTE is reduced with the increase of ε, and it tends to almost the same value in the stochastic and deterministic models when ε is relatively large. This result reveals that noise is favorable for the extinction of tumor cells under immune surveillance, and noise is ineffective when the ability of immune system is strong enough.
Conclusion
This paper is concerned with a stochastic tumorimmune system competition model. Firstly, the KuznetsovTaylor model and the Galash simplified model [9] describing the competition between the tumor and immune cells are introduced, Gaussian white noise is employed to mimic the environmental fluctuations. Then, with the methods of Itô’s formula and Lyapunov function, sufficient conditions for extinction, stochastic persistence, and strong persistence in the mean of tumor cells are established by rigorous mathematical proofs. It is shown as follows:

(A)
If \(\alpha \delta <\varepsilon \), then the effector cells \(x(t)\) go to \(\frac{\varepsilon }{\delta }\), and the tumor cells \(y(t)\) will go to extinction a.s.

(B)
If \(\alpha \delta \frac{\omega \alpha }{4\beta }>\varepsilon \), then the tumor cells \(y(t)\) will be stochastically permanent a.s.

(C)
If \(\alpha \delta >\varepsilon \), then the tumor cells \(y(t)\) will be strongly persistent in the mean a.s.
Our work reveals some important and interesting biological results. By comparing the results of Galach [9] with our work of Theorem 2, we can see that environmental noise can change the properties of tumorimmune population dynamics significantly. For example, under the same conditions, Figure 4(b) shows that the mean time to extinction of the stochastic model is less than the time in the deterministic model. Namely, environmental noise can accelerate the extinction of tumor cells under the surveillance of effector cells, which means that noise is favorable for the extinction of tumor in this condition.
Some interesting questions deserve further investigation. For example, in our model, we assume that fluctuations in the environment will mainly affect the immune coefficient. It is interesting to study what happens if noise affects other parameters of the system. Another question of interest is to consider the stability in distribution (e.g., [38, 39]). Moreover, one may propose some realistic but complex models. An example is to add the treatment into the system. The motivation is to predict the effect of treatment and design the optimal treatment schedule.
References
 1.
Cristini, V, Lowengrub, J: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach. Cambridge University Press, Cambridge (2010)
 2.
Fory, U: Marchuk’s model of immune system dynamics with application to tumor growth. J. Theor. Med. 4, 8593 (2002)
 3.
Kirschner, DE, Panetta, JC: Modeling immunotherapy of the tumorimmune interaction. J. Math. Biol. 37, 235252 (1998)
 4.
Mayer, H, Zaenker, KS, et al.: A basic mathematical model of the immune response. Chaos 5, 155161 (1995)
 5.
Waniewski, J, Zhivkov, P: A simple mathematical model for tumor immune system interactions. In: The Eight National Conference on Application of Mathematics in Biology and Medicine, pp. 149154 (2002)
 6.
Adam, JA, Bellomo, N: A Survey of Models for TumorImmune System Dynamics. Birkhauser, Boston (1997)
 7.
Eftimie, R, Bramson, JL, Earn, DJD: Interactions between the immune system and cancer: a brief review of nonspatial mathematical models. Bull. Math. Biol. 73, 232 (2011)
 8.
Kuznetsov, VA, Taylor, MA, et al.: Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bull. Math. Biol. 2, 295321 (1994)
 9.
Galach, M: Dynamics of the tumorimmune system competition  the effect of time delay. Int. J. Appl. Math. Comput. Sci. 13, 395406 (2003)
 10.
Forni, G, Giovarelli, M, et al.: Cytokine induced tumor immunogeneticity: from exogenous cytokines to gene therapy. J. Immunother. 14, 253257 (1993)
 11.
Greller, LD, Tobin, FL, Poste, G: Tumor heterogenity and progression: conceptual foundation for modelin. Invasion Metastasis 16, 177208 (1996)
 12.
Burger, R, Barton, NH: The Mathematical Theory of Selection, Recombination, and Mutation. Wiley, New York (2000)
 13.
Dieckmann, O, Heesterbeek, JAP: Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley, New York (2000)
 14.
May, R, Nowak, MA: Virus Dynamic: Mathematical Principles of Immunology and Virology. Oxford University Press, Oxford (2000)
 15.
Perelson, A, Weisbuch, G: Immunology for physicists. Rev. Mod. Phys. 69, 12191268 (1997)
 16.
Zhao, Y, Jiang, D, O’Regan, D: The extinction and persistence of the stochastic sis epidemic model with vaccination. Physica A 392, 49164927 (2013)
 17.
Wang, Z, Xu, Y, Yang, H: Levy noise induced stochastic resonance in an fhn model. Sci. China, Technol. Sci. 59, 371375 (2016)
 18.
Xu, Y, Li, Y, Zhang, H, Li, X, Kurths, J: The switch in a genetic toggle system with Levy noise. Sci. Rep. 6, 31505 (2016)
 19.
Xu, Y, Feng, J, Li, J, Zhang, H: Levy noise induced switch in the gene transcriptional regulatory system. Chaos 23, 013110 (2013)
 20.
Xu, Y, Wu, J, Du, L, Yang, H: Stochastic resonance in a genetic toggle model with harmonic excitation and Levy noise. Chaos Solitons Fractals 92, 91100 (2016)
 21.
Xu, Y, et al.: Levy noiseinduced stochastic resonance in a bistable system. Eur. Phys. J. B 86, 198 (2013)
 22.
Xu, Y, Li, Y, et al.: The phase transition in a bistable Duffing system driven by Levy noise. J. Stat. Phys. 158, 120131 (2015)
 23.
Li, Y, Xu, Y, Kurthsand, J, Yue, X: Levy noise induced transport in a rough triplewell potential. Phys. Rev. E 94, 042222 (2016)
 24.
Xu, Y, Gu, R, Zhang, H: Effects of random noise in a dynamical model of love. Chaos Solitons Fractals 44, 490497 (2011)
 25.
Xu, Y, Wang, H, Liu, D, Huang, H: Sliding mode control of a class of fractional chaotic systems in the presence of parameter perturbations. J. Vib. Control 21, 435448 (2015)
 26.
Zhong, W, Shao, Y, He, Z: Pure multiplicative stochastic resonance of a theoretical antitumor model with seasonal modulability. Phys. Rev. E 73, 95 (2006)
 27.
Albano, G, Giorno, V: A stochastic model in tumor growth. J. Theor. Biol. 242, 329336 (2006)
 28.
Lenbury, Y, Triampo, W, Tang, IM, et al.: A stochastic model of cancer growth with immune response. J. Korean Phys. Soc. 49, 16521666 (2006)
 29.
Ferrante, L, Bompadre, S, et al.: Parameter estimation in a Gompertzian stochastic model for tumor growth. Biometrics 56, 10761081 (2006)
 30.
Xu, Y, Feng, J, Li, J, Zhang, H: Stochastic bifurcation for a tumorimmune system with symmetric Levy noise. Physica A 392, 47394748 (2013)
 31.
Fiasconaro, A, Spagnolo, B: Cooccurrence of resonant activation and noiseenhanced stability in a model of cancer growth in the presence of immune response. Phys. Rev. E 74, 159 (2006)
 32.
Liu, M, Fan, M: Permanence of stochastic LotkaVolterra systems. J. Nonlinear Sci. (2016). doi:10.1007/s0033201693372
 33.
Liu, M, Wang, K: Persistence and extinction in stochastic nonautonomous logistic systems. J. Math. Anal. Appl. 375, 443457 (2011)
 34.
Mao, X, Marion, G: Environmental Brownian noise suppresses explosions in populations dynamics. Stoch. Process. Appl. 97, 95110 (2002)
 35.
Evans, LC: An Introduction to Stochastic Differential Equations. Am. Math. Soc., New York (2013)
 36.
Mao, X: Stochastic Differential Equations and Applications. Horwood, Chichester (1997)
 37.
Higham, DJ: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525546 (2001)
 38.
Liu, M, Bai, C: Dynamics of a stochastic oneprey twopredator model with Levy jumps. Appl. Math. Comput. 284, 308321 (2016)
 39.
Liu, M, Bai, C: Optimal harvesting of a stochastic mutualism model with Levy jumps. Appl. Math. Comput. 276, 301309 (2016)
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11402157 and 11571009), Shanxi Scholarship Council of China (Grant No. 2015032), Technological Innovation Programs of Higher Education Institutions in Shanxi (Grant No. 2015121) and Applied Basic Research Programs of Shanxi Province (Grant No. 2016021013).
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Authors’ contributions
YL proposed the model and drafted the main part of the manuscript. DL contributed to the revision of the manuscript and polished the language. All authors read and approved the final manuscript.
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Keywords
 tumorimmune system competition model
 environmental noise
 extinction
 persistence